Grashof number

Free convection is caused by a change in density of a fluid due to a temperature change or gradient. Usually the density decreases due to an increase in temperature and causes the fluid to rise. This motion is caused by the buoyancy force. The major force that resists the motion is the viscous force. The Grashof number is a way to quantify the opposing forces.[3]

The Grashof number is:

${\displaystyle \mathrm {Gr} _{L}={\frac {g\beta (T_{s}-T_{\infty })L^{3}}{\nu ^{2}}}\,}$ for vertical flat plates

${\displaystyle \mathrm {Gr} _{D}={\frac {g\beta (T_{s}-T_{\infty })D^{3}}{\nu ^{2}}}\,}$ for pipes

${\displaystyle \mathrm {Gr} _{D}={\frac {g\beta (T_{s}-T_{\infty })D^{3}}{\nu ^{2}}}\,}$ for bluff bodies

where:

• g is gravitational acceleration due to Earth
• β is the coefficient of volume expansion (equal to approximately 1/T for ideal gases)
• T_s is the surface temperature
• T_∞ is the bulk temperature
• L is the vertical length
• D is the diameter
• ν is the kinematic viscosity.

The L and D subscripts indicate the length scale basis for the Grashof number.

The transition to turbulent flow occurs in the range 10⁸ < Gr_L < 10⁹ for natural convection from vertical flat plates. At higher Grashof numbers, the boundary layer is turbulent; at lower Grashof numbers, the boundary layer is laminar, that is, in the range 10³ < Gr_L < 10⁶.

There is an analogous form of the Grashof number used in cases of natural convection mass transfer problems. In the case of mass transfer, natural convection is caused by concentration gradients rather than temperature gradients.[2]

$${\displaystyle \mathrm {Gr} _{c}={\frac {g\beta ^{*}(C_{a,s}-C_{a,a})L^{3}}{\nu ^{2}}}}$$

where

$${\displaystyle \beta ^{*}=-{\frac {1}{\rho }}\left({\frac {\partial \rho }{\partial C_{a}}}\right)_{T,p}}$$

and:

• g is gravitational acceleration due to Earth
• C_a,s is the concentration of species a at surface
• C_a,a is the concentration of species a in ambient medium
• L is the characteristic length
• ν is the kinematic viscosity
• ρ is the fluid density
• C_a is the concentration of species a
• T is the temperature (constant)
• p is the pressure (constant).

This discussion involving the energy equation is with respect to rotationally symmetric flow. This analysis will take into consideration the effect of gravitational acceleration on flow and heat transfer. The mathematical equations to follow apply both to rotational symmetric flow as well as two-dimensional planar flow.

$${\displaystyle {\frac {\partial }{\partial s}}(\rho ur_{0}^{n})+{\frac {\partial }{\partial y}}(\rho vr_{0}^{n})=0}$$

where:

• ${\displaystyle s}$ is the rotational direction, i.e. direction parallel to the surface
• ${\displaystyle u}$ is the tangential velocity, i.e. velocity parallel to the surface
• ${\displaystyle y}$ is the planar direction, i.e. direction normal to the surface
• ${\displaystyle v}$ is the normal velocity, i.e. velocity normal to the surface
• ${\displaystyle r_{0}}$ is the radius.

In this equation the superscript n is to differentiate between rotationally symmetric flow from planar flow. The following characteristics of this equation hold true.

• ${\displaystyle n}$ = 1: rotationally symmetric flow
• ${\displaystyle n}$ = 0: planar, two-dimensional flow
• ${\displaystyle g}$ is gravitational acceleration

This equation expands to the following with the addition of physical fluid properties:

$${\displaystyle \rho \left(u{\frac {\partial u}{\partial s}}+v{\frac {\partial u}{\partial y}}\right)={\frac {\partial }{\partial y}}\left(\mu {\frac {\partial u}{\partial y}}\right)-{\frac {dp}{ds}}+\rho g.}$$

From here we can further simplify the momentum equation by setting the bulk fluid velocity to 0 (${\displaystyle u=0}$).

$${\displaystyle {\frac {dp}{ds}}=\rho _{0}g}$$

This relation shows that the pressure gradient is simply a product of the bulk fluid density and the gravitational acceleration. The next step is to plug in the pressure gradient into the momentum equation.

$${\displaystyle \rho \left(u{\frac {\partial u}{\partial s}}+v{\frac {\partial u}{\partial y}}\right)=\mu \left({\frac {\partial ^{2}u}{\partial y^{2}}}\right)+\rho g\beta (T-T_{0})}$$

Further simplification of the momentum equation comes by substituting the volume expansion coefficient, density relationship ${\displaystyle \rho _{0}-\rho =\beta \rho (T-T_{0})}$, found above, and kinematic viscosity relationship, ${\displaystyle \nu ={\frac {\mu }{\rho }}}$, into the momentum equation.

$${\displaystyle u\left({\frac {\partial u}{\partial s}}\right)+v\left({\frac {\partial v}{\partial y}}\right)=\nu \left({\frac {\partial ^{2}u}{\partial y^{2}}}\right)+g\beta (T-T_{0})}$$

To find the Grashof number from this point, the preceding equation must be non-dimensionalized. This means that every variable in the equation should have no dimension and should instead be a ratio characteristic to the geometry and setup of the problem. This is done by dividing each variable by corresponding constant quantities. Lengths are divided by a characteristic length, ${\displaystyle L_{c}}$. Velocities are divided by appropriate reference velocities, ${\displaystyle V}$, which, considering the Reynolds number, gives ${\displaystyle V={\frac {\mathrm {Re} _{L}\nu }{L_{c}}}}$. Temperatures are divided by the appropriate temperature difference, ${\displaystyle (T_{s}-T_{0})}$. These dimensionless parameters look like the following:

• ${\displaystyle s^{*}={\frac {s}{L_{c}}}}$,
• ${\displaystyle y^{*}={\frac {y}{L_{c}}}}$,
• ${\displaystyle u^{*}={\frac {u}{V}}}$,
• ${\displaystyle v^{*}={\frac {v}{V}}}$, and
• ${\displaystyle T^{*}={\frac {(T-T_{0})}{(T_{s}-T_{0})}}}$.

The asterisks represent dimensionless parameter. Combining these dimensionless equations with the momentum equations gives the following simplified equation.

${\displaystyle u^{*}{\frac {\partial u^{*}}{\partial s^{*}}}+v^{*}{\frac {\partial u^{*}}{\partial y^{*}}}=\left[{\frac {g\beta (T_{s}-T_{0})L_{c}^{3}}{\nu ^{2}}}\right]{\frac {T^{*}}{\mathrm {Re} _{L}^{2}}}+{\frac {1}{\mathrm {Re} _{L}}}{\frac {\partial ^{2}u^{*}}{\partial {y^{*}}^{2}}}}$

where:

${\displaystyle T_{s}}$ is the surface temperature

${\displaystyle T_{0}}$ is the bulk fluid temperature

${\displaystyle L_{c}}$ is the characteristic length.

The dimensionless parameter enclosed in the brackets in the preceding equation is known as the Grashof number:

${\displaystyle \mathrm {Gr} ={\frac {g\beta (T_{s}-T_{0})L_{c}^{3}}{\nu ^{2}}}.}$

Another form of dimensional analysis that will result in the Grashof number is known as the Buckingham π theorem. This method takes into account the buoyancy force per unit volume, ${\displaystyle F_{b}}$ due to the density difference in the boundary layer and the bulk fluid.

$${\displaystyle F_{b}=(\rho -\rho _{0})g}$$

This equation can be manipulated to give,

$${\displaystyle F_{b}=-\beta g\rho _{0}\Delta T.}$$

The list of variables that are used in the Buckingham π method is listed below, along with their symbols and dimensions.

Variable	Symbol	Dimensions
Significant length	${\displaystyle L}$	${\displaystyle \mathrm {L} }$
Fluid viscosity	${\displaystyle \mu }$	${\displaystyle \mathrm {\frac {M}{Lt}} }$
Fluid heat capacity	${\displaystyle c_{p}}$	${\displaystyle \mathrm {\frac {Q}{MT}} }$
Fluid thermal conductivity	${\displaystyle k}$	${\displaystyle \mathrm {\frac {Q}{LtT}} }$
Volume expansion coefficient	${\displaystyle \beta }$	${\displaystyle \mathrm {\frac {1}{T}} }$
Gravitational acceleration	${\displaystyle g}$	${\displaystyle \mathrm {\frac {L}{t^{2}}} }$
Temperature difference	${\displaystyle \Delta T}$	${\displaystyle \mathrm {T} }$
Heat transfer coefficient	${\displaystyle h}$	${\displaystyle \mathrm {\frac {Q}{L^{2}tT}} }$

With reference to the Buckingham π theorem there are 9 – 5 = 4 dimensionless groups. Choose L, ${\displaystyle \mu ,}$ k, g and ${\displaystyle \beta }$ as the reference variables. Thus the ${\displaystyle \pi }$ groups are as follows:

${\displaystyle \pi _{1}=L^{a}\mu ^{b}k^{c}\beta ^{d}g^{e}c_{p}}$,

${\displaystyle \pi _{2}=L^{f}\mu ^{g}k^{h}\beta ^{i}g^{j}\rho }$,

${\displaystyle \pi _{3}=L^{k}\mu ^{l}k^{m}\beta ^{n}g^{o}\Delta T}$,

${\displaystyle \pi _{4}=L^{q}\mu ^{r}k^{s}\beta ^{t}g^{u}h}$.

Solving these ${\displaystyle \pi }$ groups gives:

${\displaystyle \pi _{1}={\frac {\mu (c_{p})}{k}}=\mathrm {Pr} }$,

${\displaystyle \pi _{2}={\frac {l^{3}g\rho ^{2}}{\mu ^{2}}}}$,

${\displaystyle \pi _{3}=\beta \Delta T}$,

${\displaystyle \pi _{4}={\frac {hL}{k}}=\mathrm {Nu} }$

From the two groups ${\displaystyle \pi _{2}}$ and ${\displaystyle \pi _{3},}$ the product forms the Grashof number:

${\displaystyle \pi _{2}\pi _{3}={\frac {\beta g\rho ^{2}\Delta TL^{3}}{\mu ^{2}}}=\mathrm {Gr} .}$

Taking ${\displaystyle \nu ={\frac {\mu }{\rho }}}$ and ${\displaystyle \Delta T=(T_{s}-T_{0})}$ the preceding equation can be rendered as the same result from deriving the Grashof number from the energy equation.

${\displaystyle \mathrm {Gr} ={\frac {\beta g\Delta TL^{3}}{\nu ^{2}}}}$

In forced convection the Reynolds number governs the fluid flow. But, in natural convection the Grashof number is the dimensionless parameter that governs the fluid flow. Using the energy equation and the buoyant force combined with dimensional analysis provides two different ways to derive the Grashof number.

It is also possible to derive the Grashof number by physical definition of the number as follows:

$${\displaystyle \mathrm {Gr} ={\frac {\mathrm {Buoyancy~Force} }{\mathrm {Friction~Force} }}={\frac {mg}{\tau A}}={\frac {L^{3}\rho \beta (\Delta T)g}{\mu (V/L)L^{2}}}={\frac {L^{2}\beta (\Delta T)g}{\nu V}}}$$

However, above expression, especially the final part at the right hand side, is slightly different from Grashof number appearing in literature. Following dimensionally correct scale in terms of dynamic viscosity can be used to have the final form.

$${\displaystyle \mathrm {\mu } =\rho VL}$$

Writing above scale in Gr gives;

$${\displaystyle \mathrm {Gr} ={\frac {L^{3}\beta (\Delta T)g}{\nu ^{2}}}}$$

Physical reasoning is helpful to grasp the meaning of the number. On the other hand, following velocity definition can be used as a characteristic velocity value for making certain velocities nondimensional.

$${\displaystyle \mathrm {V} ={\frac {L^{2}\beta (\Delta T)g}{\nu Gr}}}$$